Projects: I will outline some projects for you to work in small groups of size 4 or 5. These projects will be assigned to students on a first-come-first-serve basis.
I will continuously update these projects and will also add links and references to useful resources. I find these projects incredibly beautiful, interesting, and fascinating. They are easily accessible (they take very little background) and will quickly take you on the road to research. The projects listed below need only basic techniques from elementary number theory, elementary Linear algebra, Euclidean geometry, and elementary probability theory. If you have not taken any of these courses, that is fine. I will teach you or provide you with the necessary background material as and when needed.
The 2^k-numbers game over integers and finite groups: The reference for this topic is "Roots to Research" Chapter 1 (all sections). The Chapter talks in great detail about the Four numbers game which can be described as follows: start with a unit square with four integers at the four corners. In the first step, form a new
square connecting the midpoints of the four sides of the given
square. The numbers attached to the four corners of this new
square are the absolute differences of the corresponding corners
of the original square. For example, if the first square has
numbers (2, 6, 1, 9), then the next square has numbers (4, 5, 8,
7). In the following interations of the game we get (1, 3, 1, 3),
(2, 2, 2, 2), and (0, 0, 0 , 0). Thus starting with any square, we
play the game until we reach a square with numbers (0, 0, 0,
0). Several questions arise very naturally here. When is a game
finite? What integers can be lengths of games? How does the length
of a game change under permutation of the numbers of the original
square? etc. In the same way one can also play an n-numbers game
starting with a regular n-gon. It can be shown that every
n-numbers games has finite length if and only if n is a power of
2, say 2^k. Therefore the 2^k-numbers deserve special
attention. The goal of the project is to investigate the theory of
2^k-numbers games after reading the 4=2^2-numbers game theory
given in Chapter 1. So far, all this discussion is over the
integers. In another direction we shall investigate, starting with
the 2^2 case, these games with elements in some finite
groups. Here I expect to see some pleasant surprises!
Pythagorean triples and the integer points on a Hyperboloid: The reference for this topic is "Roots to Research" Chapter 2 (Sections 1, 3, and 4) and the Student Projects in The college Mathematics Journal: Vol 23, No. 5, Nov 1992 and Vol 37, No. 1, Jan 2006. The starting point is the study of positive integer solutions to the Pythagorean equation x^2 + y^2 = z^2. These are the Pythagorean triplets (a, b, c) such that a^2 + b^2 = c^2. There is endless literature on this fascinating topic. The first step is to learn some of this material which is relevant for our project; Sections 1, 3 and 4 of Chapter 2 in Roots to Research is a good place. The next step, following the guideline given in the College Math Journals (see above), we shall study the integer points on the Hyperboloid x^2 + y^2 = z^2 + k where k is a fixed positive integer. We shall study their properties, the trees, algorithms for generation etc. Lots of interesting questions will pop-up in this journey.
Pick's theorem and related problems in lattice point geometry: The reference for this topic is "Roots to Research" Chapter 3 (Sections 1, 2, 3, 5, 6, 7, 8). Pick's theorem is a rare and beautiful gem in classical mathematics. It can be stated as follows. Let P be a simple lattice polygon (i.e. a connected polygon whose vertices have integer coordinates and whose sides do not interest each other). Pick's theorem states that the area A enclosed by P is given A = I + B/2 - 1 where I is the number of lattice points in the interior of P, and B is the number of lattice points on the boundary of P. The beauty of this theorem lies in its simplicity and depth. It can be explained easily to any 4rd grader but on the other hand mathematicians are still investigating some of its deep consequences. In the project we shall investigate the possible values of B for a given value of I. We shall do this systematically beginning with triangles and parallelograms. We shall also study how Pick's theorem can be used to derive some surprising and interesting formulas for computing the gcd of two integers. Some related problems in lattice point geometry which we shall explore are questions which ask for the existence of lattice regular n-gons in 3-dimensional space.
Hamming Codes over finite groups. The reference for this is the
appendix of Abstract Algebra: An Introduction, by
Hungerford. There will be more references added later. But this is a
good book to start with. This project requires some knowledge in group
theory and Linear algebra over finite fields. If you don't have a background in these
topics I would not recommend this project. But if you enjoy abstract
algebra this would be a great thing to consider. These are some
elegant error correcting codes which are constructed using matrices
over finite fields. These are very interesting applications; see
recent posting on
my BLOG to get a
feel for the topic. The goal of the project is to construct new
Hamming-type codes using group theory. The reference for this is the
appendix of Abstract Algebra: An Introduction, by Hungerford.
Matrix number theory: I have no reference for
this project. It is just a plain and broad question of doing number
theory with numbers replaced by matrices. Every single questions
that you encounter in number theory can be investigated in the world
of matrices. For example, what are prime matrices? Which matrices
are sum of squares of matrices? and so on. Take your favourite book
on number theory and try to redo the entire number theory in the
book over matrices. Of course, one has to make some reasonable and
appropriate assumption, and figuring those out is also part of the project.
Research Teams and Abstracts
The k-Numbers Game: Let k be a positive integer. We investigate the k-Numbers game, in which k numbers are arranged at the vertices of a regular polygon with k sides and the absolute value of the difference between adjacent numbers is taken until the numbers at all vertices become 0. We investigate symmetry in the game, upper bounds for the length of games, and the effect properties of numbers have on the length of games, and related questions.
Ellen Sparks (Primary presenter)
Pythagorean Triples and Integer Points on a Hyperboloid: We begin by studying the Pythagorean equation: x^2 + y^2 = z^2. We investigate what positive integer triples (x, y, z) will satisfy the Pythagorean equation and explore the patterns in the solutions that emerge. The natural extension of this work is to consider the equation of the Hyperboloid x^2 + y^2 = z^2 + k, where k is a fixed positive integer. We study the problem of finding integer points
located on this Hyperboloid when k=2, i.e., solutions in positive integers. In particular, we investigate properties of the solutions and algorithms for generating all the integer points on the Hyperboloid.
Brian Finney (Primary presenter)
Pick's Theorem and its Applications Pick's theorem gives a nice formula to compute the area enclosed by a simple lattice polygon (a polygon whose vertices have integer coordinates). The area A of such a polygon P is given in terms of the number I of lattice points located in the interior of P and the number B of lattice points on the boundary of P as: A = I + B/2 - 1. We will investigate several consequences of this theorem.
Eric Larson (Primary presenter)
Jan 25: Register for the undergraduate research syposium by Feb 7th.
Jan 20: Log on to blackboard to read the latex assignment. You
have to upload your LaTeX file on Blackboard. Due: Feb 1st.
Jan 16: First LaTex exercise: Download Winedt to your laptops or
desktops (use the Latex link above). Then download
this SAMPLE FILE,
and open it using Winedt.(Do not change the filename or its
extension.) Finally, after you open this file in winedt, hit the
Latex button on the top and you should be able to see the output in
a .dvi file.
Jan 12: Be Wise, Generalise: I want you to generalise this to 3 dimensions: Call
a cuboid (3-dimensional rectangle) cool if one if its 3 sides is
integer length. Show that a cuboid that admits a tiling by cool
subcuboids is itself cool.
Jan 11: In your research Journal, write a complete and precise
proof of the following theorem which was discussed in class: A
rectangle is called cool if either its length of width is of integer
length. If R is a rectangle that admits a tiling by cool
subrectangles, then R has to be cool.